Lemma 5 [1]} 
Consider a nonnegative and irreducible matrix \(B=[b_{ij}]\in \mathbb {R}^{3\times 3}\)  which satisfies that \(b_{ii}<\lambda ^*\) , where \(\lambda ^*>0\)  and \(i=1,2,3\) . Then \(\rho (B) < \lambda ^*\)  iff \(\text{det}[\lambda ^*I-B]>0\) .

